Kleczka was arrested at least twice for drunk driving. A 1995 arrest for drunken driving — his blood alcohol content was twice the legal limit in Virginia — led him to acknowledge his alcoholism and seek treatment. Friends say it softened his rough edges. He turned to God as part of his recovery and continued to meet regularly with fellow recovering alcoholics.[3]

For his first 10-and-a-half terms, Kleczka represented a district that included most of the southern half of Milwaukee, as well as part of eastern Waukesha County. After the 2000 census, the 5th District, covering downtown and north Milwaukee and represented by fellow Democrat Tom Barrett, was eliminated, and most of its territory was merged with the 4th District. The new 4th was a more compact district located solely in Milwaukee County, and took in all of the city of Milwaukee; it was by far the most Democratic district in Wisconsin. While the 4th was more Barrett's district than Kleczka's, Barrett opted to run for governor, effectively handing the seat to Kleczka.

The pronunciation of Kleczka's name often proved baffling to the uninformed. Rep. Frank Annunzio (D-IL) repeatedly butchered it to the point that an exasperated Kleczka took to calling him "Rep. Annunciation."

Kleczka announced his retirement in 2004, and did not run for reelection. He officially retired in January 2005 after ten terms in Congress, and was succeeded by State Senator Gwen Moore, also a Democrat.

After Kleczka retired, he moved to Middleton, Wisconsin with his wife. Kleczka died on October 8, 2017 from natural causes at a care facility, in the Madison, Wisconsin area.[4][5][6]